Lord Rayleigh on Waves. 267 



The shortest wave-length is when /3 = I ; and then 

 2^ :Z = 5-96. 



If/3= 3' 2a>:Z=8-4. 



If/3= 8' 2^:Z = 12-6. 



These results are in agreement with Russell's observations. 



The form of the wave as determined by (J) is shown in the 

 figure, half the wave only being drawn : — 



The velocity of propagation is given by (H), which is Scott 

 Russell's formula exactly. In words, the velocity of the wave 

 is that due to half the greatest depth of the water. 



Another of Russell's observations is now readily accounted 

 for : — " It was always found that the wave broke when its ele- 

 vation above the general level became equal, or nearly so, to 

 the greatest depth. The application of mathematics to this 

 circumstance is so difficult, that we confine ourselves to the 

 mention of the observed fact"*. When the wave is treated 

 as stationary, it is evident from dynamics that its height can 

 never exceed that due to the velocity of the stream in the un- 



disturbed parts ; that is, V — lis less than _JL. But u\-=gl' ', and 



therefore V — I is less than \ V ', or V — lis less than I. When the 

 wave is on the point of breaking, the water at the crest is 

 moving with the velocity of the wave. 



Periodic Waves in Deep Water. - 



The best known theory of these waves is that of Grerstner, 

 Rankine, and Froude, in which the profile is trochoidal. The 

 motion of each particle of the fluid is in a circle, which is de^- 

 scribed with uniform velocity. If A, k be the coordinates of 

 the centre of one of these circles, measured horizontally and 

 downwards respectively, the position of the particle at time t 



* Airy, ' Tides and Waves/ art. 401. 



